The Agony and the Apostasy

The Creation of Adam, Michelangelo Buonaroti

The construction of the Sistine Chapel (Capella Sistina) was completed in 1481. Its construction was commissioned by Pope Sixtus IV for whom it was named. One hundred thirty four feet long and 44 feet wide, the huge chapel is a towering 68 feet high. The ceiling, originally painted in a field of blue with gold stars, was covered in beautiful frescoes by Michelangelo at the behest of Pope Julius II.

Michelangelo labored for four years, from 1508 to 1512. He painted over 5,000 square feet of frescoes containing over 300 figures. Some of the most beautiful art ever created, such as The Creation of Adam, embellish the chapel’s ceiling. In addition to being one of the world’s most talented sculptors and painters, Michelangelo proved to be a highly competent engineer, designing a clever scaffolding system that allowed services to be held in the chapel below as he painted above. Four years of talent, genius, and backbreaking labor resulted in one of the world’s most breathtaking works of art.

And at the end of these four year, Michelangelo approached Pope Julius and, on bended knee, kissed his ring and asked for payment. “No,” the Pope responded, “you did not do that. The chapel was funded by the Church. The paints were purchased by the Church. The laborers who erected your scaffolding were paid by the Church. You did not do that.”

A small fiction, of course, but of a spirit with the narrative presented by the Democratic left. Elizabeth Warren, Barack Obama, and, most recently, Hillary Clinton assert that entrepreneurs do not create businesses and corporations do not create jobs.

Those to the political right, Senator Cruz and friends, are aghast at such apostasy, believing that no one but entrepreneurs and corporations create businesses and jobs.

How to explain this chasm?

First, let’s try to understand what each side is actually saying. The Warren camp means to say that public spending, public infrastructures, public services, are all essential to the entrepreneur and corporations. That businesses and jobs couldn’t be created without reliance on government.

Those on the right recognize that government creates a public infrastructure. And, more importantly, it creates a system of laws that protects each citizen’s liberty and property rights. But they are just as adamant that, government or no, businesses and jobs wouldn’t exist without those who create them.

Both positions are a bit more sympathetic when viewed in a larger context. But which is right? This seems a bit of a standoff, a chicken or the egg puzzle. What came first, the jobs or the highways?

To answer this, we must delve into the tangle of Aristotelian logic and the concept of necessity and sufficiency. While logic can be quite complex (after all, it underlies all computers, the internet, Facebook, and silly cat videos), in this case it is quite straightforward.

This is something you already know. Think of the components of a grilled cheese sandwich: bread and cheese. Is it necessary to have cheese to make a grilled cheese sandwich? Obviously, yes. But is it sufficient to make a grilled cheese sandwich with only cheese? Equally obviously, no.

The cheese is necessary but not sufficient. The bread is also necessary but not sufficient. One needs both cheese and bread to prepare the grilled cheese sandwich. But to make simple toast, the bread is both necessary and sufficient.

Apply to the Sistine Chapel. The Pope is correct, Michelangelo could not have painted the ceiling if the Church had not built the chapel. But the ceiling would not have been painted so beautifully without Michelangelo (or someone of equal talent). Both conditions are necessary for the result. 

Fast forward 1,500 years. Senator Warren is correct that jobs and businesses could not (easily) be created without the infrastructure provided by government. But Senator Cruz is equally correct that jobs and businesses would not exist without entrepreneurs and corporations to create them. Just like a grilled cheese sandwich, we need both government and business. One cannot thrive without the other.

One way to view it is that government creates a canvas upon which the creative, risk-taking entrepreneur paints her vision, creating businesses and jobs as a result.

Which is a tale of caution, for those who would strangle government in the extreme risk the ability of entrepreneurs to create. But those who would smother business risk government as well. It is, after all, the taxes and fees paid by businesses and corporations and taxes paid by wage employees that support government. Without a healthy, bustling economy, from where will government funding be obtained?

Food for thought next time you see corporations and businesses taking it on the chin.  

The Hitchhiker's Guide to Mathematical Balderdash

By the time you read this, the midterm elections of 2014 will have passed. You will be either ecstatic with the peoples’ decision or deeply disappointed. In either case, it’s time to take a breather and think of more ethereal things.

The internet is fascinating. The topics which ebb and flow on social media are a revealing window on the subject of human thought, susceptibility, and superstition. Perhaps a lesson or two to learn here.

For instance, a recent post making the rounds is typical, piquing our interest and suggesting some magical properties. It proposes that our shoe size can predict our age.

One of a class of such postings, this one posits that your shoe size can predict your age as follows:

1.       Take your shoe size (whole number, round up if necessary)

2.       Multiply it by 5

3.       Add 50

4.       Multiply by 20

5.       Add 1014

6.       Subtract the year you were born

And voila, the result is your shoe size as the leftmost digits and your age as the rightmost. “It’s magic!” proclaims the post. Balderdash.

Let’s take this apart, understand it, and identify its limitations.

Our first task is to express this as a simple expression:

(SHOE x 5 + 50) x 20 + 1014 – BIRTHYEAR

Let’s try it assuming a shoe size of 9 and birth year of 1971.

(9 x 5 + 50) x 20 + 1014 - 1971

The result is 943. Shoe magic, indeed! This person’s shoe size is 9 and age 43!

Dang – how did it know?

Rest assured, there is no mystery here. Let’s apply a little math. Starting with the first expression, we can reduce and represent it as follows:

((SHOE x 5) + 50) x 20 + 1014 – BIRTHYEAR

SHOE x 100 + 1000 + 1014 – BIRTHYEAR

SHOE x 100 + 2014 – BIRTHYEAR   

                                                                             

Now it begins to make a bit more sense. SHOE x 100 shifts the shoe size to the left and 2014-BIRTHYEAR yields the person’s age. Adding them together gives us the result:  943 in this case.

A few things might become obvious to you at this point. The use of shoe size is completely arbitrary. We could use hat size or the number of cups of oatmeal in your breakfast or any other number. Shoe size, per se, has nothing at all to do with it.

If you play around, you will also find that if the birth year is over 100 years ago, the calculation breaks down. Also, once we get to next year (2015), the age calculation is no longer valid: it only works for 2014 because of the constant “1014” in the original expression.

No magic at all, this is a cheap algebraic parlor trick.

Another recent math problem making the rounds of social media raised more acrimonious debate than that of the supporters of Senator Elizabeth Warren vs. those of Senator Ted Cruz. My goodness, math has only one right answer, what is the grounds of debate?

This one is based on mistaken assumptions, perhaps a cautionary note in all of our dealings.

Take a look at this expression:

36 / 6 x 3 + 2

In other words, 36, divided by 6, times 3, plus 2. But in which order do we apply these operations?

One option is to divide 36 by six first, then multiply by 3 , finally adding 2. The answer would be 20.

18 + 2 = 20

Another alternative is to multiply 6 by 3 first, the divide into 36, finally adding 2 yielding 4.

36 / 18 + 2 = 4

Two distinct answers, 20 and 4. Which is correct? The wrong answer could blow up the next resupply mission to the International Space Station (math applied to the real world can be really important).

There is a guide known by the acronym of PEMDAS which describes the order in which operations are to be performed:

1.       Parenthesis

2.       Exponents

3.       Multiplication

4.       Division

5.       Addition

6.       Subtraction

This guide is a gentleman’s agreement meant to remove ambiguity. But the prescription leaves itself open to misinterpretation.

A common (incorrect) assumption is that multiplication comes first relative to division but, in fact, multiplication and division are of equal weight  (as are addition and subtraction). When operations are of equal weight, they are processed left to right as encountered. The correct way to interpret PEMDAS is PE(MD)(AS), meaning;

1.       Parenthesis

2.       Exponents

3.       Multiplication and division, equal weight, left to right

4.       Addition and subtraction, equal weight, left to right

So the proper way to evaluate the above expression is as follows:

36 / 6 x 3 + 2

6 x 3 + 2

18 + 2

20

If you guessed 20, you win the prize!

Enough of the numbers. Revel (or commiserate) the recent election results, and embrace a bold new confidence in debunking the Internet’s mathematical puzzles. You can do it!

Your mission - an informed vote

Governing the body civitas is the most important thing that we, in our role as citizens, do. And it is important that each of our individual decisions be well informed. Soon, Massachusetts citizens will decide on a slate of representatives and settle several ballot questions, the latter an exercise in direct democracy.

When selecting a governmental representative, you must decide which candidate best represents your interests. This is, unfortunately, often involves holding one’s nose and voting for the least bad choice (a topic for another day). But ballot questions constitute pure democracy. Here, instead of selecting a representative and trusting her or his judgment, you make the call. The only way to responsibly do so is to be well informed.

Your choices are simple. The first one is whether to vote at all.

In the recent primary election, only 16% of registered voters turned out to vote (this may account for the nose-holding factor). In the upcoming general midterm, perhaps 40% will cast a ballot. Think of what this means if you choose not to vote.

Imagine a random group of ten citizens, strangers, having coffee at Morin’s. Four of them have decided to vote in the upcoming election and six won’t bother. If you are one of the six, you are in effect telling the four “Whatever you decide, I’m fine with it. I’m putting my family’s well-being in your hands.”

That’s your right as a citizen, but it may not be optimal. Why would you give four strangers unchallenged sway over your family’s welfare? A better alternative would be to inform yourself and cast your ballot.

Ah, but there is the rub, the “inform yourself” part. For if you don’t fully understand the issues and likely outcomes, how can you vote intelligently? Voting for a poor outcome out of ignorance is, perhaps, worse than not voting at all.

The Commonwealth has recognized the importance of voter information and, in the case of ballot questions, has provided a voter guide. This guide  provides a summary of each question, the effect of a yes or no vote, and arguments written by proponents and opponents. The Commonwealth hastens to establish that these arguments are only opinions. You, dear voter, must still assess what you think to be the truth. The Commonwealth may inform, but you must decide.

Let’s take Question 1 as a case in point and attempt to wend an objective path through the countervailing opinions.

First, the facts. Facts should be identified as incontrovertible and not subject to interpretation.

• The current fuel (gasoline and diesel) tax in Massachusetts is 26.5 cents per gallon.
• Section 1 of chapter 64A of the General Laws requires that the tax per gallon be adjusted annually by the percentage change of the Consumer Price Index (from the preceding year) with a lower limit of 21.5 cents per gallon (should the CPI decline)
• Question 1 would strike the requirement that the gas tax be automatically adjusted

Let’s take a look at key points in the arguments. First, in favor:

“Voting yes simply stops the linkage of the gas tax to inflation.” Given the facts, this statement is true.

“This initiative cuts no money for bridge or road repair.” This one is a bit slippery. No, the initiative does not decrease the current 26.5 cent levy. But assuming that the CPI increases, it would reduce the funds automatically available in future years.

 Now the opposition:

“Question One threatens the safety of you and your family when traveling on Massachusetts’ roads and bridges.” This is loaded with assumptions, the foremost being that the legislature will not meet its obligation to maintain public infrastructure, whether by raising taxes, raising fees, or shifting resources to do so.

“A Yes vote would make things even worse, by taking away existing gas tax revenues that we need to solve this public safety crisis…” No, the initiative would not reduce existing tax revenues. It removes the automatic increases. The legislature is free to vote for future increases or reallocation of funds as noted.

Reading between the lines, we can summarize the two positions thusly:

The pro-Question 1 camp feels that the legislature should take affirmative action to provide funding for roads and bridges, and that each such decision should be subject to a vote. Taxation with representation is a sound policy, but there are many precedents for automatic increases. (When has your property tax remained flat year to year?)

The anti-Question 1 group thinks that the legislature will not provide adequate funding by vote and that automatic increases are the way around this problem. But if the legislature is failing, did the voters choose their representatives well?

What’s the right choice? It’s completely up to you. But this is the approach you must take to each of the questions. Inform yourself, think about it, read between the lines, then make your decision. And the miracle of democracy is that, whatever the outcome, the people generally tend to be right.

Spurious Correlations and Dr. Alice Stewart's Triumph

Shoe fitting by x-ray (fluoroscope)

Who says statistics are dry and boring? Tyler Vigen, a Harvard Law student, recently created a fun website called Spurious Correlations. Tyler pokes fun by finding sets of numbers which, by random coincidence, are related. He notes, for instance, that the total number of US Political Action Committees is strongly related to the number of people who die by falling out of a wheelchair. Or that the number of people who have died from being tangled in their bedsheets tracks the total revenue generated by US ski facilities.

Amusing nonsense. And proof positive that correlation does not prove causation. In other words, that two sets of data are correlated does not mean that one causes the other.

The word comes to us from Middle French and means "related together." Two sets of measurements or data that track each other are correlated. Statistically, a relatively simple procedure can be applied to theses sets resulting in a relationship measurement (correlation coefficient) that ranges from -1 to +1. A coefficient of 0 means that the two sets of data are completely unrelated, i.e., none of the changes in one set can be explained by changes in the other. Coefficients of 1 show perfect correlation, the sets of data change in lockstep with each other. (-1 is a perfect inverse relationship, +1 a perfect direct relationship).

Ice cream cones purchased when compared to outdoor temperature forms a strong direct relationship: as temperatures increase, purchases of ice cream likewise increase. As men increase in age, the number of hairs on their head decreases; this is an inverse relationship and, unfortunately, is strongly correlated.

People are quick to tell you that correlation does not prove causation, usually when they disagree with the findings of some study. And they are correct; as Vigen demonstrates, coincidental correlations are not rare.

But as an informed citizen, you must be aware that correlation is a powerful data analysis tool used in fields ranging from health studies to astrophysics to economics. When two sets of numbers are correlated, statisticians do not jump to the conclusion that one causes the other, but rather are put on alert that deeper analysis is called for. Correlation is not proof, but it directs the investigation.

Such as the investigation performed by Dr. Alice Mary Stewart which ultimately saved millions of children the scourge of childhood cancer.

To set the stage, you must realize that radiation and x-rays, discovered in the 1890s by Marie Curie and Wilhelm Roentgen, were considered miracles of the modern age. The danger of exposure was not fully appreciated, and radiation was used in the twentieth century for such applications as illuminating watch dials, fitting shoes, and viewing a fetus in the womb. All were considered safe.

But Dr. Stewart was on a quest to explain a rash of lymphatic leukemia and other childhood cancers in England. Her own godchild died of the disease which drove her interest.

Dr. Stewart and her team conducted a study of 203 English public health hospitals during 1953-1955, looking for details of all children who had contracted cancer. The study included a questionnaire filled out by each mother.

From this mass of data, Dr. Stewart and her statistician, George Kneale, searched for patterns, looked for correlations. And they found one. It turns out that the children of mothers who had had a fetal x-ray were twice as likely to contract leukemia as those who had not had an x-ray. The finding was stunning, and rejected. After all, that radiation was safe was settled science. And Dr. Stewart was, after all (sniff), only a woman.

But she, and Kneale, persevered. Their working relationship was interesting. Whenever Stewart proposed a conclusion, Kneale would apply all of his substantial statistical skill to prove her wrong. This was their method and it made her results stronger as she defended them against stout attacks.

It took nearly twenty years, but Stewart and Kneale amassed a growing dataset of 22,000 childhood cancer victims in which the use of pre-natal x-rays was increasingly complicit. Finally, in the late 1970s, the American and British medical societies accepted Dr. Stewart's findings and recommended that pre-natal x-rays not be routinely performed.

Two lessons here. First, correlation does not prove causation, but it is often a smoking gun begging for deeper analysis.  And the other - share your conclusions and data, welcome challenges, and defend them with logic and reason. Do not dismiss another's theory just because she doesn't share your prejudices.

And now, dear citizen, you know more of correlation than most do.